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poses_quat.py
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poses_quat.py
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import math_utility
import numpy as np
import scipy.linalg.matfuncs
import scipy.spatial.transform
import scipy.stats
import poses_euler
"""
Operations and functions related to SE(3) poses / poses pdf expressed as 7d vector quaternion + 3d
Notes
-----
Implementation based on [1]. The last updated version of the paper can be found here :
https://github.com/jlblancoc/tutorial-se3-manifold
Pose pdf are represented are dict with keys "pose_mean" and "pose_cov".
Point pdf are are represented are dict with keys "mean" and "cov".
References
----------
.. [1] "J.-L. Blanco. A tutorial on se (3) transformation parameterizations and
on-manifold optimization. University of Malaga, Tech. Rep, 3, 2010." (Last update : 11/05/2020)
.. [2] "Y. Breux, MpIC_oldVersion_noApprox" (in doc folder)
.. [3] "Breux, Yohan, André Mas, and Lionel Lapierre.
On-manifold Probabilistic ICP: Application to Underwater Karst Exploration." (2021)
"""
def fromqrxyzToqxyzr(q):
"""
Convert quaternion format from [qr qx qy qz] (MRPT) to [qx qy qz qr] (numpy)
Parameters
----------
q : array
quaternion in format [qr qx qy qz]
Returns
-------
array
quaternion in format [qx qy qz qr]
"""
return np.array([q[1], q[2], q[3], q[0]])
def fromqrxyzToqxyzr_array(q):
"""
Convert array of quaternions with format from [qr qx qy qz] (MRPT) to [qx qy qz qr] (numpy)
Parameters
----------
q : 2D array
array of quaternions in format [qr qx qy qz]
Returns
-------
2D array
array of quaternions in format [qx qy qz qr]
"""
res = np.empty((q.shape[0],4))
res[:, 0] = q[:, 1]
res[:, 1] = q[:, 2]
res[:, 2] = q[:, 3]
res[:, 3] = q[:, 0]
return res
def fromqxyzrToqrxyz(q):
"""
Convert quaternion format from [qx qy qz qr] (numpy) to [qr qx qy qz] (MRPT)
Parameters
----------
q : array
quaternion in format [qx qy qz qr]
Returns
-------
array
quaternion in format [qr qx qy qz]
"""
return np.array([q[3], q[0], q[1], q[2]])
def fromqxyzrToqrxyz_array(q):
"""
Convert array of quaternions with format from [qx qy qz qr] (MRPT) to [qr qx qy qz] (numpy)
Parameters
----------
q : 2D array
array of quaternions in format [qx qy qz qr]
Returns
-------
2D array
array of quaternions in format [qr qx qy qz]
"""
res = np.empty((q.shape[0],4))
res[:,0] = q[:,3]
res[:,1] = q[:,0]
res[:,2] = q[:,1]
res[:,3] = q[:,2]
return res
def composePoseQuat(q1, q2):
"""
Compose two poses expressed as quaternion + 3d
Parameters
----------
q1 : array
first pose
q2 : array
second pose
Returns
-------
array
composed pose q1 + q2
Notes
-----
See [1], section 5.2.1
"""
# TODO : implement without using rotation matrix
quat_arr_1 = fromqrxyzToqxyzr(q1)
quat_arr_2 = fromqrxyzToqxyzr(q2)
rot1 = scipy.spatial.transform.Rotation.from_quat(quat_arr_1)
rot2 = scipy.spatial.transform.Rotation.from_quat(quat_arr_2)
t1 = q1[4:7]
t2 = q2[4:7]
rot_12 = scipy.spatial.transform.Rotation.from_matrix(np.matmul(rot1.as_matrix(), rot2.as_matrix()))
t_12 = rot1.apply(t2) + t1
quat_res = rot_12.as_quat()
return np.block([fromqxyzrToqrxyz(quat_res), t_12])
def composePoseQuat_array(q1, q2):
"""
Compose an array of poses with a pose (in quaternion + 3d)
Parameters
----------
q1 : 2D array or array
array of first poses
q2 : array or 2D array
second poses
Returns
-------
2D array
array of poses q1 + q2 for each q1 in q1 or each q2 in q2
Notes
-----
See [1], section 5.2.1
"""
if(q1.ndim == 2):
res = np.empty((q1.shape[0],7))
quat_arr_1 = fromqrxyzToqxyzr_array(q1)
quat_arr_2 = fromqrxyzToqxyzr(q2)
rot1 = scipy.spatial.transform.Rotation.from_quat(quat_arr_1)
rot2 = scipy.spatial.transform.Rotation.from_quat(quat_arr_2)
t1_array = q1[:,4:7]
t2 = q2[4:7]
rot_12_array = scipy.spatial.transform.Rotation.from_matrix(np.einsum('...ij,jk->...ik', rot1.as_matrix(), rot2.as_matrix()))
res[:,0:4] = fromqxyzrToqrxyz_array(rot_12_array.as_quat())
res[:,4:7] = np.einsum('...ij,j->...i', rot1.as_matrix(),t2) + t1_array
elif (q2.ndim == 2):
res = np.empty((q2.shape[0], 7))
quat_arr_1 = fromqrxyzToqxyzr(q1)
quat_arr_2 = fromqrxyzToqxyzr_array(q2)
rot1 = scipy.spatial.transform.Rotation.from_quat(quat_arr_1)
rot2 = scipy.spatial.transform.Rotation.from_quat(quat_arr_2)
t1 = q1[4:7]
t2 = q2[:,4:7]
rot_12_array = scipy.spatial.transform.Rotation.from_matrix(np.einsum('ij,kjl->kil', rot1.as_matrix(), rot2.as_matrix()))
res[:, 0:4] = fromqxyzrToqrxyz_array(rot_12_array.as_quat())
res[:, 4:7] = rot1.apply(t2) + t1
return res
def composePoseQuatPoint(q_poseQuat, x):
"""
Compose a pose and point in quaternion + 3d
Parameters
----------
q_poseQuat : array
pose
x : array
3D point
Returns
-------
array
composed point q_poseQuat + x
Notes
-----
See [1], section 3.2.1
"""
qx_sqr = q_poseQuat[1]**2
qy_sqr = q_poseQuat[2]**2
qz_sqr = q_poseQuat[3]**2
qr_qx = q_poseQuat[0] * q_poseQuat[1]
qr_qy = q_poseQuat[0] * q_poseQuat[2]
qr_qz = q_poseQuat[0] * q_poseQuat[3]
qx_qy = q_poseQuat[1] * q_poseQuat[2]
qx_qz = q_poseQuat[1] * q_poseQuat[3]
qy_qz = q_poseQuat[2] * q_poseQuat[3]
ax = x[0]
ay = x[1]
az = x[2]
point_composed = [ q_poseQuat[4] + ax + 2.*(-(qy_sqr + qz_sqr)*ax + (qx_qy - qr_qz)*ay + (qr_qy + qx_qz)*az),
q_poseQuat[5] + ay + 2.*( (qr_qz + qx_qy)*ax - (qx_sqr + qz_sqr)*ay + (qy_qz - qr_qx)*az),
q_poseQuat[6] + az + 2.*( (qx_qz - qr_qy)*ax + (qr_qx + qy_qz)*ay - (qx_sqr + qy_sqr)*az )
]
return point_composed
def inversePoseQuat_point(q_poseQuat, point):
"""
Point relative to the pose (Inverse compose pose-point) in quaternion + 3d
Parameters
----------
q_poseQuat : array
pose
point : array
3D point
Returns
-------
array
Coords of "point" relative to "q_poseQuat"
Notes
-----
See [1], section 4.2.1
"""
qx_sqr = q_poseQuat[1]**2
qy_sqr = q_poseQuat[2]**2
qz_sqr = q_poseQuat[3]**2
qr_qx = q_poseQuat[0] * q_poseQuat[1]
qr_qy = q_poseQuat[0] * q_poseQuat[2]
qr_qz = q_poseQuat[0] * q_poseQuat[3]
qx_qy = q_poseQuat[1] * q_poseQuat[2]
qx_qz = q_poseQuat[1] * q_poseQuat[3]
qy_qz = q_poseQuat[2] * q_poseQuat[3]
dx = point[0] - q_poseQuat[4]
dy = point[1] - q_poseQuat[5]
dz = point[2] - q_poseQuat[6]
a_composed = np.array([ dx + 2.*( -(qy_sqr + qz_sqr)*dx + (qx_qy + qr_qz)*dy + (-qr_qy + qx_qz)*dz ),
dy + 2.*( (-qr_qz + qx_qy)*dx - (qx_sqr + qz_sqr)*dy + (qy_qz + qr_qx)*dz ),
dz + 2.*( (qx_qz + qr_qy)*dx + (-qr_qx + qy_qz)*dy - (qx_sqr + qy_sqr)*dz )])
return a_composed
def inversePoseQuat_point_array(q_poseQuat_array, point):
"""
Point relative to an array of poses (Inverse compose pose-point) in quaternion + 3d
Parameters
----------
q_poseQuat_array : 2D array
array of poses
point : array
3D point
Returns
-------
2D array
Array of coords of "point" relative to each pose in "q_poseQuat_array"
Notes
-----
See [1], section 4.2.1
"""
qx_sqr = np.power(q_poseQuat_array[:,1],2)
qy_sqr = np.power(q_poseQuat_array[:,2],2)
qz_sqr = np.power(q_poseQuat_array[:,3],2)
qr_qx = q_poseQuat_array[:,0] * q_poseQuat_array[:,1]
qr_qy = q_poseQuat_array[:,0] * q_poseQuat_array[:,2]
qr_qz = q_poseQuat_array[:,0] * q_poseQuat_array[:,3]
qx_qy = q_poseQuat_array[:,1] * q_poseQuat_array[:,2]
qx_qz = q_poseQuat_array[:,1] * q_poseQuat_array[:,3]
qy_qz = q_poseQuat_array[:,2] * q_poseQuat_array[:,3]
dx = point[0] - q_poseQuat_array[:,4]
dy = point[1] - q_poseQuat_array[:,5]
dz = point[2] - q_poseQuat_array[:,6]
a_composed = np.empty((q_poseQuat_array.shape[0],3))
a_composed[:,0] = dx + 2.*( -(qy_sqr + qz_sqr)*dx + (qx_qy + qr_qz)*dy + (-qr_qy + qx_qz)*dz )
a_composed[:,1] = dy + 2.*( (-qr_qz + qx_qy)*dx - (qx_sqr + qz_sqr)*dy + (qy_qz + qr_qx)*dz )
a_composed[:,2] = dz + 2.*( (qx_qz + qr_qy)*dx + (-qr_qx + qy_qz)*dy - (qx_sqr + qy_sqr)*dz )
return a_composed
def inversePoseQuat(q_poseQuat):
"""
Inverse of a pose in quaternion + 3d
Parameters
----------
q_poseQuat: array
pose
Returns
-------
array
Inversed pose
Notes
-----
See [1], section 6.2.1
"""
t = inversePoseQuat_point(q_poseQuat,[0.,0.,0.])
return np.block([np.array([q_poseQuat[0], -q_poseQuat[1], -q_poseQuat[2], -q_poseQuat[3]]), t])
def inversePoseQuat_array(q_poseQuat_array):
"""
Inverse of an array of poses in quaternion + 3d
Parameters
----------
q_poseQuat_array : 2D array
array of poses
Returns
-------
2D array
array of inversed poses
Notes
-----
See [1], section 6.2.1
"""
t_array = inversePoseQuat_point_array(q_poseQuat_array, [0.,0.,0.])
res = np.empty((q_poseQuat_array.shape[0],4))
res[:,0] = q_poseQuat_array[:,0]
res[:,1] = -q_poseQuat_array[:,1]
res[:,2] = -q_poseQuat_array[:,2]
res[:,3] = -q_poseQuat_array[:,3]
res[:,4] = t_array
return res
def jacobianQuatNormalization(quat):
"""
Jacobian of quaternion normalization
Parameters
----------
quat : array
quaternion or quaternion + 3d pose
Returns
-------
2D array
4 X 4 Jacobian matrix
Notes
-----
See [1], section 1.2.2 equation (1.7)
"""
qr_sqr = quat[0] ** 2
qx_sqr = quat[1] ** 2
qy_sqr = quat[2] ** 2
qz_sqr = quat[3] ** 2
qx_qr = quat[1] * quat[0]
qy_qr = quat[2] * quat[0]
qz_qr = quat[3] * quat[0]
qx_qy = quat[1] * quat[2]
qx_qz = quat[1] * quat[3]
qy_qz = quat[2] * quat[3]
K = 1./np.linalg.norm(quat[0:4])**3
jacobian_normalization = np.array([ [qx_sqr + qy_sqr + qz_sqr, -qx_qr, -qy_qr, -qz_qr],
[-qx_qr, qr_sqr + qy_sqr + qz_sqr, -qx_qy, -qx_qz],
[-qy_qr, -qx_qy, qr_sqr + qx_sqr + qz_sqr, -qy_qz],
[-qz_qr, -qx_qz, -qy_qz, qr_sqr + qx_sqr + qy_sqr]
])
return K*jacobian_normalization
''' Jacobian of quaternion normalization (for an array of quaternions) '''
def jacobianQuatNormalization_array(quat_array):
"""
Jacobians of quaternion normalization evaluated at an array of quaternions
Parameters
----------
quat_array : 2D array
array of quaternions or quaternion + 3d poses
Returns
-------
3D array
array of 4 X 4 jacobian matrices
Notes
-----
Each slice [i,:,:] corresponds to jacobian evaluated at the i-th quaternion in the input
See [1], section 1.2.2 equation (1.7)
"""
qr_sqr = np.power(quat_array[:,0], 2)
qx_sqr = np.power(quat_array[:,1], 2)
qy_sqr = np.power(quat_array[:,2], 2)
qz_sqr = np.power(quat_array[:,3], 2)
qx_qr = quat_array[:,1] * quat_array[:,0]
qy_qr = quat_array[:,2] * quat_array[:,0]
qz_qr = quat_array[:,3] * quat_array[:,0]
qx_qy = quat_array[:,1] * quat_array[:,2]
qx_qz = quat_array[:,1] * quat_array[:,3]
qy_qz = quat_array[:,2] * quat_array[:,3]
K = 1./np.power(np.linalg.norm(quat_array[:,0:4],axis=1), 3)
jacobian_normalization = np.empty((quat_array.shape[0],4,4))
jacobian_normalization[:,0,0] = qx_sqr + qy_sqr + qz_sqr
jacobian_normalization[:,0,1] = -qx_qr
jacobian_normalization[:,0,2] = -qy_qr
jacobian_normalization[:,0,3] = -qz_qr
jacobian_normalization[:,1,0] = -qx_qr
jacobian_normalization[:,1,1] = qr_sqr + qy_sqr + qz_sqr
jacobian_normalization[:,1,2] = -qx_qy
jacobian_normalization[:,1,3] = -qx_qz
jacobian_normalization[:,2,0] = -qy_qr
jacobian_normalization[:,2,1] = -qx_qy
jacobian_normalization[:,2,2] = qr_sqr + qx_sqr + qz_sqr
jacobian_normalization[:,2,3] = -qy_qz
jacobian_normalization[:,3,0] = -qz_qr
jacobian_normalization[:,3,1] = -qx_qz
jacobian_normalization[:,3,2] = -qy_qz
jacobian_normalization[:,3,3] = qr_sqr + qx_sqr + qy_sqr
return K[:,None,None]*jacobian_normalization
def computeJacobianQuat_composePosePoint(quat, a):
"""
Jacobian of pose-point composition in quaternion + 3d
Parameters
----------
quat : array
quaternion or quaternion + 3d pose
a : array
3D point
Returns
-------
jacobianQuat_composePosePoint_pose : 2D array
3 X 4 jacobian matrix wrt the pose
jacobianQuat_composePosePoint_point : 2D array
3 X 3 jacobian matrix wrt the point
Notes
-----
See [1], section 3.2.2
"""
qr_ax = quat[0]*a[0]
qr_ay = quat[0]*a[1]
qr_az = quat[0]*a[2]
qx_ax = quat[1] * a[0]
qx_ay = quat[1] * a[1]
qx_az = quat[1] * a[2]
qy_ax = quat[2] * a[0]
qy_ay = quat[2] * a[1]
qy_az = quat[2] * a[2]
qz_ax = quat[3] * a[0]
qz_ay = quat[3] * a[1]
qz_az = quat[3] * a[2]
A = np.array([[-qz_ay + qy_az, qy_ay + qz_az, -2.*qy_ax + qx_ay + qr_az, -2.*qz_ax - qr_ay + qx_az],
[qz_ax - qx_az, qy_ax - 2.*qx_ay - qr_az, qx_ax + qz_az, qr_ax -2.*qz_ay + qy_az],
[-qy_ax + qx_ay, qz_ax + qr_ay - 2.*qx_az, -qr_ax + qz_ay - 2.*qy_az, qx_ax + qy_ay]
])
qr_sqr = quat[0]**2
qx_sqr = quat[1]**2
qy_sqr = quat[2]**2
qz_sqr = quat[3]**2
qx_qr = quat[1]*quat[0]
qy_qr = quat[2]*quat[0]
qz_qr = quat[3]*quat[0]
qx_qy = quat[1]*quat[2]
qx_qz = quat[1]*quat[3]
qy_qz = quat[2]*quat[3]
K = 1./np.linalg.norm(quat[0:4])**3
jacobian_normalization = K*np.array([ [qx_sqr + qy_sqr + qz_sqr, -qx_qr, -qy_qr, -qz_qr],
[-qx_qr, qr_sqr + qy_sqr + qz_sqr, -qx_qy, -qx_qz],
[-qy_qr, -qx_qy, qr_sqr + qx_sqr + qz_sqr, -qy_qz],
[-qz_qr, -qx_qz, -qy_qz, qr_sqr + qx_sqr + qy_sqr]
])
jacobianQuat_composePosePoint_pose_quat = 2.*np.matmul(A,jacobian_normalization)
jacobianQuat_composePosePoint_pose = np.block([jacobianQuat_composePosePoint_pose_quat,np.eye(3)])
jacobianQuat_composePosePoint_point = 2.*np.array([ [0.5 - qy_sqr - qz_sqr, qx_qy - qz_qr, qy_qr + qx_qz],
[qz_qr + qx_qy, 0.5 - qx_sqr - qz_sqr, qy_qz - qx_qr],
[qx_qz - qy_qr, qx_qr + qy_qz, 0.5 - qx_sqr - qy_sqr]
])
return jacobianQuat_composePosePoint_pose, jacobianQuat_composePosePoint_point
def computeJacobianQuat_composePosePoint_quatArray(quat_array, a):
"""
Jacobian of pose-point composition evaluated at an array of quaternion in quaternion + 3d
Parameters
----------
quat_array : 2D array
array of quaternions or quaternion + 3d poses
a : array
3D point
Returns
-------
jacobianQuat_composePosePoint_pose : 3D array
array of 3 X 4 jacobian matrix wrt each quat/poses in quat_array
jacobianQuat_composePosePoint_point : 3D array
array of 3 X 3 jacobian matrix wrt the point for each quat/poses in quat_array
Notes
-----
See [1], section 3.2.2
"""
qr_ax = quat_array[:,0] * a[0]
qr_ay = quat_array[:,0] * a[1]
qr_az = quat_array[:,0] * a[2]
qx_ax = quat_array[:,1] * a[0]
qx_ay = quat_array[:,1] * a[1]
qx_az = quat_array[:,1] * a[2]
qy_ax = quat_array[:,2] * a[0]
qy_ay = quat_array[:,2] * a[1]
qy_az = quat_array[:,2] * a[2]
qz_ax = quat_array[:,3] * a[0]
qz_ay = quat_array[:,3] * a[1]
qz_az = quat_array[:,3] * a[2]
A = np.empty((quat_array.shape[0],3,4))
A[:,0,0] = -qz_ay + qy_az
A[:,0,1] = qy_ay + qz_az
A[:,0,2] = -2.*qy_ax + qx_ay + qr_az
A[:,0,3] = -2.*qz_ax - qr_ay + qx_az
A[:,1,0] = qz_ax - qx_az
A[:,1,1] = qy_ax - 2.*qx_ay - qr_az
A[:,1,2] = qx_ax + qz_az
A[:,1,3] = qr_ax -2.*qz_ay + qy_az
A[:,2,0] = -qy_ax + qx_ay
A[:,2,1] = qz_ax + qr_ay - 2.*qx_az
A[:,2,2] = -qr_ax + qz_ay - 2.*qy_az
A[:,2,3] = qx_ax + qy_ay
qr_sqr = np.power(quat_array[:,0], 2)
qx_sqr = np.power(quat_array[:,1],2)
qy_sqr = np.power(quat_array[:,2],2)
qz_sqr = np.power(quat_array[:,3],2)
qx_qr = quat_array[:,1]*quat_array[:,0]
qy_qr = quat_array[:,2]*quat_array[:,0]
qz_qr = quat_array[:,3]*quat_array[:,0]
qx_qy = quat_array[:,1]*quat_array[:,2]
qx_qz = quat_array[:,1]*quat_array[:,3]
qy_qz = quat_array[:,2]*quat_array[:,3]
K = 1. / np.power(linalg.norm(quat_array[:,0:4]), 3)
jacobian_normalization = np.empty((quat_array.shape[0], 4, 4))
jacobian_normalization[:, 0, 0] = qx_sqr + qy_sqr + qz_sqr
jacobian_normalization[:, 0, 1] = -qx_qr
jacobian_normalization[:, 0, 2] = -qy_qr
jacobian_normalization[:, 0, 3] = -qz_qr
jacobian_normalization[:, 1, 0] = -qx_qr
jacobian_normalization[:, 1, 1] = qr_sqr + qy_sqr + qz_sqr
jacobian_normalization[:, 1, 2] = -qx_qy
jacobian_normalization[:, 1, 3] = -qx_qz
jacobian_normalization[:, 2, 0] = -qy_qr
jacobian_normalization[:, 2, 1] = -qx_qy
jacobian_normalization[:, 2, 2] = qr_sqr + qx_sqr + qz_sqr
jacobian_normalization[:, 2, 3] = -qy_qz
jacobian_normalization[:, 3, 0] = -qz_qr
jacobian_normalization[:, 3, 1] = -qx_qz
jacobian_normalization[:, 3, 2] = -qy_qz
jacobian_normalization[:, 3, 3] = qr_sqr + qx_sqr + qy_sqr
jacobian_normalization = K[:,None,None]*jacobian_normalization
jacobianQuat_composePosePoint_pose_quat_array = 2.*np.einsum('kij,kjl->kil',A,jacobian_normalization)
jacobianQuat_composePosePoint_pose_array = np.empty((quat_array.shape[0],3,7))
jacobianQuat_composePosePoint_pose_array[:,:,0:4] = jacobianQuat_composePosePoint_pose_quat_array
ones = np.full(quat_array.shape[0],1.)
jacobianQuat_composePosePoint_pose_array[:,0,4] = ones
jacobianQuat_composePosePoint_pose_array[:,1,5] = ones
jacobianQuat_composePosePoint_pose_array[:,2,6] = ones
jacobianQuat_composePosePoint_point_array = np.empyt(quat_array.shape[0],3,3)
jacobianQuat_composePosePoint_point_array[:,0,0] = 0.5 - qy_sqr - qz_sqr,
jacobianQuat_composePosePoint_point_array[:,0,1] = qx_qy - qz_qr
jacobianQuat_composePosePoint_point_array[:,0,2] = qy_qr + qx_qz
jacobianQuat_composePosePoint_point_array[:,1,0] = qz_qr + qx_qy
jacobianQuat_composePosePoint_point_array[:,1,1] = 0.5 - qx_sqr - qz_sqr
jacobianQuat_composePosePoint_point_array[:,1,2] = qy_qz - qx_qr
jacobianQuat_composePosePoint_point_array[:,2,0] = qx_qz - qy_qr
jacobianQuat_composePosePoint_point_array[:,2,1] = qx_qr + qy_qz
jacobianQuat_composePosePoint_point_array[:,2,2] = 0.5 - qx_sqr - qy_sqr
return jacobianQuat_composePosePoint_pose_array, 2.*jacobianQuat_composePosePoint_point
def computeJacobianQuat_composePosePoint_pointArray(quat, a_array):
"""
Jacobian of pose-point composition evaluated at an array of points in quaternion + 3d
Parameters
----------
quat : array
quaternion or quaternion + 3d pose
a_array : 2D array
array of 3D points
Returns
-------
jacobianQuat_composePosePoint_pose : 3D array
array of 3 X 4 jacobian matrix wrt the pose for each point in a_array
jacobianQuat_composePosePoint_point : 3D array
array of 3 X 3 jacobian matrix wrt the point for each point in a_array
Notes
-----
See [1], section 3.2.2
"""
qr_ax = quat[0] * a_array[:,0]
qr_ay = quat[0] * a_array[:,1]
qr_az = quat[0] * a_array[:,2]
qx_ax = quat[1] * a_array[:,0]
qx_ay = quat[1] * a_array[:,1]
qx_az = quat[1] * a_array[:,2]
qy_ax = quat[2] * a_array[:,0]
qy_ay = quat[2] * a_array[:,1]
qy_az = quat[2] * a_array[:,2]
qz_ax = quat[3] * a_array[:,0]
qz_ay = quat[3] * a_array[:,1]
qz_az = quat[3] * a_array[:,2]
A = np.empty((a_array.shape[0], 3, 4))
A[:, 0, 0] = -qz_ay + qy_az
A[:, 0, 1] = qy_ay + qz_az
A[:, 0, 2] = -2. * qy_ax + qx_ay + qr_az
A[:, 0, 3] = -2. * qz_ax - qr_ay + qx_az
A[:, 1, 0] = qz_ax - qx_az
A[:, 1, 1] = qy_ax - 2. * qx_ay - qr_az
A[:, 1, 2] = qx_ax + qz_az
A[:, 1, 3] = qr_ax - 2. * qz_ay + qy_az
A[:, 2, 0] = -qy_ax + qx_ay
A[:, 2, 1] = qz_ax + qr_ay - 2. * qx_az
A[:, 2, 2] = -qr_ax + qz_ay - 2. * qy_az
A[:, 2, 3] = qx_ax + qy_ay
qr_sqr = np.power(quat[0], 2)
qx_sqr = np.power(quat[1], 2)
qy_sqr = np.power(quat[2], 2)
qz_sqr = np.power(quat[3], 2)
qx_qr = quat[1] * quat[0]
qy_qr = quat[2] * quat[0]
qz_qr = quat[3] * quat[0]
qx_qy = quat[1] * quat[2]
qx_qz = quat[1] * quat[3]
qy_qz = quat[2] * quat[3]
K = 1. / np.linalg.norm(quat[0:4]) ** 3
jacobian_normalization = K * np.array([[qx_sqr + qy_sqr + qz_sqr, -qx_qr, -qy_qr, -qz_qr],
[-qx_qr, qr_sqr + qy_sqr + qz_sqr, -qx_qy, -qx_qz],
[-qy_qr, -qx_qy, qr_sqr + qx_sqr + qz_sqr, -qy_qz],
[-qz_qr, -qx_qz, -qy_qz, qr_sqr + qx_sqr + qy_sqr]
])
jacobianQuat_composePosePoint_pose_quat_array = 2. * np.einsum('kij,jl->kil', A, jacobian_normalization)
jacobianQuat_composePosePoint_pose_array = np.zeros((a_array.shape[0], 3, 7))
jacobianQuat_composePosePoint_pose_array[:, :, 0:4] = jacobianQuat_composePosePoint_pose_quat_array
ones = np.full(a_array.shape[0], 1.)
jacobianQuat_composePosePoint_pose_array[:, 0, 4] = ones
jacobianQuat_composePosePoint_pose_array[:, 1, 5] = ones
jacobianQuat_composePosePoint_pose_array[:, 2, 6] = ones
jacobianQuat_composePosePoint_point = 2. * np.array([[0.5 - qy_sqr - qz_sqr, qx_qy - qz_qr, qy_qr + qx_qz],
[qz_qr + qx_qy, 0.5 - qx_sqr - qz_sqr, qy_qz - qx_qr],
[qx_qz - qy_qr, qx_qr + qy_qz, 0.5 - qx_sqr - qy_sqr]
])
return jacobianQuat_composePosePoint_pose_array, jacobianQuat_composePosePoint_point
''' J_q+a -> Jacobian of pose-point composition in quaternion (for q an array of quaternions) '''
def computeJacobianQuat_composePosePoint_array(quat, a):
"""
Jacobian of pose-point composition evaluated at an array of quaternions/poses or an array of points
Parameters
----------
quat : array or 2D array
quaternion/pose or array of quaternions/poses
a : array or 2D array
3D point or array of 3D points
Returns
-------
jacobian_pose : 3D array
jacobians relative to the pose
jacobian_point : 3D array
jacobians relative to the point
Notes
-----
Only one of the input should be an array.
See [1], section 3.2.2
"""
if(quat.ndim == 2):
return computeJacobianQuat_composePosePoint_quatArray(quat, a)
else:
return computeJacobianQuat_composePosePoint_pointArray(quat, a)
def computeJacobianQuat_composePose(q_poseQuat1, q_poseQuat2):
"""
Jacobian of pose composition in quaternion + 3d
Parameters
----------
q_poseQuat1 : array
first pose
q_poseQuat2 : array
second pose
Returns
-------
jacobian_q1 : 2D array
7 X 7Jacobian matrix wrt the first pose
jacobian_q2 : 2D array
7 X 7 Jacobian matrix wrt the second pose
Notes
-----
See [1], section 5.2.2
"""
q_poseQuat_compose = composePoseQuat(q_poseQuat1,q_poseQuat2)
jacobianQuat_composePosePoint_pose, jacobianQuat_composePosePoint_point = computeJacobianQuat_composePosePoint(q_poseQuat1, q_poseQuat2[4:7])
J_normalization = jacobianQuatNormalization(q_poseQuat_compose)
# Jacobian of the quaternion part w.r.t to quaternion variables
jacobian_quat_quat_q1 = np.array([ [q_poseQuat2[0] , -q_poseQuat2[1] , -q_poseQuat2[2] , -q_poseQuat2[3]],
[q_poseQuat2[1] , q_poseQuat2[0] , q_poseQuat2[3] , -q_poseQuat2[2]],
[q_poseQuat2[2] , -q_poseQuat2[3] , q_poseQuat2[0] , q_poseQuat2[1]] ,
[q_poseQuat2[3] , q_poseQuat2[2] , -q_poseQuat2[1] , q_poseQuat2[0]]
])
jacobian_quat_quat_q1 = np.matmul(J_normalization, jacobian_quat_quat_q1)
jacobian_quat_quat_q2 = np.array([ [q_poseQuat1[0] , -q_poseQuat1[1] , -q_poseQuat1[2] , -q_poseQuat1[3]],
[q_poseQuat1[1] , q_poseQuat1[0] , -q_poseQuat1[3] , q_poseQuat1[2]] ,
[q_poseQuat1[2] , q_poseQuat1[3] , q_poseQuat1[0] , -q_poseQuat1[1]],
[q_poseQuat1[3] , -q_poseQuat1[2] , q_poseQuat1[1] , q_poseQuat1[0]]
])
jacobian_quat_quat_q2 = np.matmul(J_normalization, jacobian_quat_quat_q2)
jacobian_q1 = np.block([[jacobian_quat_quat_q1, np.zeros((4,3))],
[jacobianQuat_composePosePoint_pose],
])
jacobian_q2 = np.block([[jacobian_quat_quat_q2, np.zeros((4,3))],
[np.zeros((3, 4)), jacobianQuat_composePosePoint_point]
])
return jacobian_q1, jacobian_q2
def computeJacobianQuat_composePose_firsPoseArray(q_poseQuat1_array, q_poseQuat2):
"""
Jacobian of pose position evaluated at an array of first poses
Parameters
----------
q_poseQuat1_array : 2D array
array of first poses
q_poseQuat2 : array
second pose
Returns
-------
jacobian_q1 : 3D array
array of 7 X 7 jacobian matrix wrt first pose
jacobian_q2 : 3D array
array of 7 X 7 jacobian matrix wrt second pose
Notes
-----
See [1], section 5.2.2
"""
jacobianQuat_composePosePoint_pose, jacobianQuat_composePosePoint_point = computeJacobianQuat_composePosePoint_quatArray(q_poseQuat1_array, q_poseQuat2[4:7])
#Jacobian of the quaternion part w.r.t to quaternion variables
jacobian_quat_quat_q1 = np.array([ [q_poseQuat2[0] , -q_poseQuat2[1] , -q_poseQuat2[2] , -q_poseQuat2[3]],
[q_poseQuat2[1] , q_poseQuat2[0] , q_poseQuat2[3] , -q_poseQuat2[2]],
[q_poseQuat2[2] , -q_poseQuat2[3] , q_poseQuat2[0] , q_poseQuat2[1]] ,
[q_poseQuat2[3] , q_poseQuat2[2] , -q_poseQuat2[1] , q_poseQuat2[0]]
])
jacobian_quat_quat_q1_array = np.eisum('kij,jl->kil', J_normalization, jacobian_quat_quat_q1, optimize=True)
jacobian_quat_quat_q2_array = np.empty((q_poseQuat1_array.shape[0],4,4))
jacobian_quat_quat_q2_array[:,0,0] = q_poseQuat1_array[:,0]
jacobian_quat_quat_q2_array[:,0,1] = -q_poseQuat1_array[:,1]
jacobian_quat_quat_q2_array[:,0,2] = -q_poseQuat1_array[:,2]
jacobian_quat_quat_q2_array[:,0,3] = -q_poseQuat1_array[:,3]
jacobian_quat_quat_q2_array[:,1,0] = q_poseQuat1_array[:,1]
jacobian_quat_quat_q2_array[:,1,1] = q_poseQuat1_array[:,0]
jacobian_quat_quat_q2_array[:,1,2] = -q_poseQuat1_array[:,3]
jacobian_quat_quat_q2_array[:,1,3] = q_poseQuat1_array[:,2]
jacobian_quat_quat_q2_array[:,2,0] = q_poseQuat1_array[:,2]
jacobian_quat_quat_q2_array[:,2,1] = q_poseQuat1_array[:,3]
jacobian_quat_quat_q2_array[:,2,2] = q_poseQuat1_array[:,0]
jacobian_quat_quat_q2_array[:,2,3] = -q_poseQuat1_array[:,1]
jacobian_quat_quat_q2_array[:,3,0] = q_poseQuat1_array[:,3]
jacobian_quat_quat_q2_array[:,3,1] = q_poseQuat1_array[:,2]
jacobian_quat_quat_q2_array[:,3,2] = -q_poseQuat1_array[:,1]
jacobian_quat_quat_q2_array[:,3,3] = q_poseQuat1_array[:,0]
jacobian_quat_quat_q2_array = np.einsum('kij,kjl->kil', J_normalization_array, jacobian_quat_quat_q2_array)
jacobian_q1 = np.zeros((q_poseQuat1_array.shape[0],7,7))
jacobian_q1[:,4,4] = jacobian_quat_quat_q1_array
jacobian_q1[:,3,7] = jacobianQuat_composePosePoint_pose
jacobian_q2 = np.zeros((q_poseQuat1_array.shape[0],7,7))
jacobian_q2[:,4,4] = jacobian_quat_quat_q2_array
jacobian_q2[:,3,3] = jacobianQuat_composePosePoint_point
return jacobian_q1, jacobian_q2
def computeJacobianQuat_composePose_secondPoseArray(q_poseQuat1, q_poseQuat2_array):
"""
Jacobian of pose position evaluated at an array of second poses
Parameters
----------
q_poseQuat1 : array
first pose
q_poseQuat2_array : 2D array
array of second poses
Returns
-------
jacobian_q1 : 3D array
array of 7 X 7 jacobian matrix wrt first pose
jacobian_q2 : 3D array
array of 7 X 7 jacobian matrix wrt second pose
Notes
-----
See [1], section 5.2.2
"""
jacobianQuat_composePosePoint_pose, jacobianQuat_composePosePoint_point = computeJacobianQuat_composePosePoint_pointArray(q_poseQuat1, q_poseQuat2_array[:,4:7])
# Jacobian of the quaternion part w.r.t to quaternion variables
jacobian_quat_quat_q1 = np.empty((q_poseQuat2_array.shape[0],4,4))
jacobian_quat_quat_q1[:,0,0] = q_poseQuat2_array[:,0]
jacobian_quat_quat_q1[:,0,1] = -q_poseQuat2_array[:,1]
jacobian_quat_quat_q1[:,0,2] = -q_poseQuat2_array[:,2]
jacobian_quat_quat_q1[:,0,3] = -q_poseQuat2_array[:,3]
jacobian_quat_quat_q1[:,1,0] = q_poseQuat2_array[:,1]
jacobian_quat_quat_q1[:,1,1] = q_poseQuat2_array[:,0]
jacobian_quat_quat_q1[:,1,2] = q_poseQuat2_array[:,3]
jacobian_quat_quat_q1[:,1,3] = -q_poseQuat2_array[:,2]
jacobian_quat_quat_q1[:,2,0] = q_poseQuat2_array[:,2]
jacobian_quat_quat_q1[:,2,1] = -q_poseQuat2_array[:,3]
jacobian_quat_quat_q1[:,2,2] = q_poseQuat2_array[:,0]
jacobian_quat_quat_q1[:,2,3] = q_poseQuat2_array[:,1]
jacobian_quat_quat_q1[:,3,0] = q_poseQuat2_array[:,3]
jacobian_quat_quat_q1[:,3,1] = q_poseQuat2_array[:,2]
jacobian_quat_quat_q1[:,3,2] = -q_poseQuat2_array[:,1]
jacobian_quat_quat_q1[:,3,3] = q_poseQuat2_array[:,0]
jacobian_quat_quat_q1= np.einsum('kij,kjl->kil', J_normalization_array, jacobian_quat_quat_q1, optimize=True)
jacobian_quat_quat_q2 = np.array([[q_poseQuat1[0], -q_poseQuat1[1], -q_poseQuat1[2], -q_poseQuat1[3]],
[q_poseQuat1[1], q_poseQuat1[0], -q_poseQuat1[3], q_poseQuat1[2]],
[q_poseQuat1[2], q_poseQuat1[3], q_poseQuat1[0], -q_poseQuat1[1]],
[q_poseQuat1[3], -q_poseQuat1[2], q_poseQuat1[1], q_poseQuat1[0]]
])
jacobian_quat_quat_q2 = np.einsum('kij,jl->kil', J_normalization_array, jacobian_quat_quat_q2, optimize=True)
jacobian_q1 = np.zeros((q_poseQuat2_array.shape[0], 7, 7))
jacobian_q1[:, 0:4, 0:4] = jacobian_quat_quat_q1
jacobian_q1[:, 4:7, 0:7] = jacobianQuat_composePosePoint_pose
jacobian_q2 = np.zeros((q_poseQuat2_array.shape[0], 7, 7))
jacobian_q2[:, 0:4, 0:4] = jacobian_quat_quat_q2
for k in range(0,q_poseQuat2.shape[0]):
jacobian_q2[k, 4:7, 4:7] = jacobianQuat_composePosePoint_point
return jacobian_q1, jacobian_q2
def computeJacobianQuat_composePose_array(q_poseQuat1, q_poseQuat2):
"""
Jacobian of pose position evaluated at an array of first poses OR an array of second poses
Parameters
----------
q_poseQuat1 : array or 2D array
first pose or array of 2D poses
q_poseQuat2_array : array or 2D array
second pose or array of second poses
Returns
-------
jacobian_q1 : 3D array
array of 7 X 7 jacobian matrix wrt first pose
jacobian_q2 : 3D array
array of 7 X 7 jacobian matrix wrt second pose
Notes
-----
Only one input should be an array.
See [1], section 5.2.2
"""
q_poseQuat_compose = composePoseQuat_array(q_poseQuat1, q_poseQuat2)
J_normalization_array = jacobianQuatNormalization_array(q_poseQuat_compose)
if(q_poseQuat1.ndim == 2):
return computeJacobianQuat_composePose_firstPoseArray(q_poseQuat1, q_poseQuat2[4:7])
else :
return computeJacobianQuat_composePose_secondPoseArray(q_poseQuat1, q_poseQuat2)
def computeJacobian_inversePoseQuatPoint_pose(q_poseQuat, point):
"""
Jacobian of point relative to pose wrt pose
Parameters
----------
q_poseQuat : array
pose
point : array
3D point
Returns
-------
2D array
3 X 7 jacobian matrix wrt poses
Notes
-----
See [1], section 4.2.2
"""
qx_sqr = q_poseQuat[1]**2
qy_sqr = q_poseQuat[2]**2
qz_sqr = q_poseQuat[3]**2
qr_qx = q_poseQuat[0] * q_poseQuat[1]
qr_qy = q_poseQuat[0] * q_poseQuat[2]
qr_qz = q_poseQuat[0] * q_poseQuat[3]
qx_qy = q_poseQuat[1] * q_poseQuat[2]
qx_qz = q_poseQuat[1] * q_poseQuat[3]
qy_qz = q_poseQuat[2] * q_poseQuat[3]
dx = point[0] - q_poseQuat[4]
dy = point[1] - q_poseQuat[5]
dz = point[2] - q_poseQuat[6]
qr_dx = q_poseQuat[0] * dx
qr_dy = q_poseQuat[0] * dy
qr_dz = q_poseQuat[0] * dz
qx_dx = q_poseQuat[1] * dx
qx_dy = q_poseQuat[1] * dy
qx_dz = q_poseQuat[1] * dz
qy_dx = q_poseQuat[2] * dx
qy_dy = q_poseQuat[2] * dy
qy_dz = q_poseQuat[2] * dz
qz_dx = q_poseQuat[3] * dx
qz_dy = q_poseQuat[3] * dy
qz_dz = q_poseQuat[3] * dz
J_trans = 2.*np.array([[qy_sqr + qz_sqr - 0.5, -(qr_qz + qx_qy), qr_qy - qx_qz],
[qr_qz - qx_qy, qx_sqr + qz_sqr - 0.5, -(qr_qx + qy_qz)],